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37
Determinant maximization with linear matrix inequality constraints
 SIAM Journal on Matrix Analysis and Applications
, 1998
"... constraints ..."
Error Estimations For Indirect Measurements: Randomized Vs. Deterministic Algorithms For "BlackBox" Programs
 Handbook on Randomized Computing, Kluwer, 2001
, 2000
"... In many reallife situations, it is very difficult or even impossible to directly measure the quantity y in which we are interested: e.g., we cannot directly measure a distance to a distant galaxy or the amount of oil in a given well. Since we cannot measure such quantities directly, we can measure ..."
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Cited by 29 (13 self)
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In many reallife situations, it is very difficult or even impossible to directly measure the quantity y in which we are interested: e.g., we cannot directly measure a distance to a distant galaxy or the amount of oil in a given well. Since we cannot measure such quantities directly, we can measure them indirectly: by first measuring some relating quantities x1 ; : : : ; xn , and then by using the known relation between x i and y to reconstruct the value of the desired quantity y. In practice, it is often very important to estimate the error of the resulting indirect measurement. In this paper, we describe and compare different deterministic and randomized algorithms for solving this problem in the situation when a program for transforming the estimates e x1 ; : : : ; e xn for x i into an estimate for y is only available as a black box (with no source code at hand). We consider this problem in two settings: statistical, when measurements errors \Deltax i = e x i \Gamma x i are inde...
Astrogeometry, Error Estimation, and Other Applications of SetValued Analysis
 ACM SIGNUM Newsletter
, 1996
"... In many reallife application problems, we are interested in numbers, namely, in the numerical values of the physical quantities. There are, however, at least two classes of problems, in which we are actually interested in sets: ffl In image processing (e.g., in astronomy), the desired blackand ..."
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Cited by 27 (26 self)
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In many reallife application problems, we are interested in numbers, namely, in the numerical values of the physical quantities. There are, however, at least two classes of problems, in which we are actually interested in sets: ffl In image processing (e.g., in astronomy), the desired blackandwhite image is, from the mathematical viewpoint, a set.
Setmembership filtering and a setmembership normalized LMS algorithm with an adaptive step size
 IEEE Signal Process. Lett
, 1998
"... Abstract — Setmembership identification (SMI) theory is extended to the more general problem of linearinparameters filtering by defining a setmembership specification, as opposed to a bounded noise assumption. This sets the framework for several important filtering problems that are not modeled ..."
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Cited by 25 (6 self)
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Abstract — Setmembership identification (SMI) theory is extended to the more general problem of linearinparameters filtering by defining a setmembership specification, as opposed to a bounded noise assumption. This sets the framework for several important filtering problems that are not modeled by a “true ” unknown system with bounded noise, such as adaptive equalization, to exploit the unique advantages of SMI algorithms. A recursive solution for set membership filtering is derived that resembles a variable step size normalized least mean squares (NLMS) algorithm. Interesting properties of the algorithm, such as asymptotic cessation of updates and monotonically nonincreasing parameter error, are established. Simulations show significant performance improvement in varied environments with a greatly reduced number of updates. I.
Setmembership adaptive equalization and an updatorshared implementation for multiple channel communication systems
 IEEE Trans. Signal Processing
, 1998
"... Abstract — This paper considers the problems of channel estimation and adaptive equalization in the novel framework of setmembership parameter estimation. Channel estimation using a class of setmembership identification algorithms known as optimal bounding ellipsoid (OBE) algorithms and their exte ..."
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Cited by 16 (9 self)
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Abstract — This paper considers the problems of channel estimation and adaptive equalization in the novel framework of setmembership parameter estimation. Channel estimation using a class of setmembership identification algorithms known as optimal bounding ellipsoid (OBE) algorithms and their extension to track timevarying channels are described. Simulation results show that the OBE channel estimators outperform the leastmeansquare (LMS) algorithm and perform comparably with the RLS and the Kalman filter. The concept of setmembership equalization is introduced along with the notion of a feasible equalizer. Necessary and sufficient conditions are derived for the existence of feasible equalizers in the case of linear equalization for a linear FIR additive noise channel. An adaptive OBE algorithm is shown to provide a set of estimated feasible equalizers. The selective update feature of the OBE algorithms is exploited to devise an updatorshared scheme in a multiple channel environment, referred to as updatorshared parallel adaptive equalization (USHAPE). USHAPE is shown to reduce hardware complexity significantly. Procedures to compute the minimum number of updating processors required for a specified quality of service are presented. I.
Robust Filtering And Feedforward Control Based On Probabilistic Descriptions Of Model Errors
 Automatica
, 1992
"... A new approach to robust estimation of signals, prediction of timeseries and robust feedforward control is considered. Signal and system parameter deviations are represented as random variables, with known covariances. A robust design is obtained by minimizing the squared estimation error, average ..."
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Cited by 13 (6 self)
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A new approach to robust estimation of signals, prediction of timeseries and robust feedforward control is considered. Signal and system parameter deviations are represented as random variables, with known covariances. A robust design is obtained by minimizing the squared estimation error, averaged both with respect to model errors and the noise. A polynomial equations approach, based on averaged spectral factorizations and averaged Diophantine equations, is derived. Mild solvability conditions guarantee the existence of stable optimal filters and feedforward regulators. The robust design turns out to be no more complicated than the design of an ordinary Wiener filter or LQG regulator. The proposed approach avoids two drawbacks of robust minimax design. First, probabilistic descriptions of model uncertainties may have soft bounds. These are more readily obtainable in a noisy environment than the hard bounds required for minimax design. Furthermore, not only the range of uncertainties...
Ellipsoidal bounds for uncertain linear equations and dynamical systems
 Automatica
, 2004
"... In this paper, we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations. The proposed technique is based on the combination of a quadratic embedding of the uncertainty, and the Sprocedure. This formulat ..."
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Cited by 11 (0 self)
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In this paper, we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations. The proposed technique is based on the combination of a quadratic embedding of the uncertainty, and the Sprocedure. This formulation leads to convex optimization problems that can be essentially solved in O(n 3)—n being the size of unknown vector — by means of suitable interior point barrier methods, as well as to closed form results in some particular cases. We further show that the uncertain linear equations paradigm can be directly applied to various statebounding problems for dynamical systems subject to setvalued noise and model uncertainty.
An Ellipsoid Calculus Based on Propagation and Fusion
"... This paper presents an Ellipsoidal Calculus based solely on two basic operations: propagation and fusion. Propagation refers to the problem of obtaining an ellipsoid that must satisfy an affine relation with another ellipsoid, and fusion to that of computing the ellipsoid that tightly bounds the int ..."
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Cited by 10 (5 self)
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This paper presents an Ellipsoidal Calculus based solely on two basic operations: propagation and fusion. Propagation refers to the problem of obtaining an ellipsoid that must satisfy an affine relation with another ellipsoid, and fusion to that of computing the ellipsoid that tightly bounds the intersection of two given ellipsoids. These two operations supersede the Minkowski sum and difference, affine transformation and intersection tight bounding of ellipsoids on which other ellipsoidal calculi are based. Actually, a Minkowski operation can be seen as a fusion followed by a propagation and an affine transformation as a particular case of propagation. Moreover, the presented formulation is numerically stable in the sense that it is immune to degen eracies of the involved ellipsoids and/or affine relations. Examples arising
Tracking of TimeVarying Parameters using Optimal Bounding Ellipsoid Algorithms
 Proc., 34th Annual Allerton Conf. Communication, Control and Computing, University of Illinois, UrbanaChampaign, Oct 24
, 1996
"... This paper analyzes the performance of an optimal bounding ellipsoid (OBE) algorithm for tracking timevarying parameters with incrementally bounded time variations. A linear statespace model is used, with the timevarying parameters represented by the state vector. The OBE algorithm exhibits a sel ..."
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Cited by 6 (5 self)
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This paper analyzes the performance of an optimal bounding ellipsoid (OBE) algorithm for tracking timevarying parameters with incrementally bounded time variations. A linear statespace model is used, with the timevarying parameters represented by the state vector. The OBE algorithm exhibits a selective update property for the time and observationupdate equations, and necessary and sufficient conditions for state tracking are derived. The interpretability of the optimization criterion is also investigated along with simulation results. 1 Introduction Tracking of time varying parameters is an important problem, both from theoretical as well as practical viewpoints, in adaptive signal processing, communication and control systems. An elegant, convenient and general framework for formulating the problem is provided by linear statespace equations. In this paper, we use the discretetime state equation framework and present an optimal bounding ellipsoid (OBE) algorithm for tracking tim...
SetMembership State Estimation with Optimal Bounding Ellipsoids
 Proc. Intl. Symp. on Information Theory and its App
, 1996
"... This paper presents a setmembership state estimation scheme for linear systems with unknown but bounded inputs and noise. The approach is to compute 100% confidence regions for the state vector in the form of optimal ellipsoidal sets in the statespace. The proposed algorithm entails greatly reduced ..."
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Cited by 5 (5 self)
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This paper presents a setmembership state estimation scheme for linear systems with unknown but bounded inputs and noise. The approach is to compute 100% confidence regions for the state vector in the form of optimal ellipsoidal sets in the statespace. The proposed algorithm entails greatly reduced computational complexity in comparison to other state estimation schemes (e.g., Schweppe's algorithm [1, 2] and the celebrated Kalman filter [3, 4]) due to a significant reduction in the order of the matrix inversion involved. As a result, the new algorithm can be expected to circumvent the wellknown problem of numerical instability observed in the Kalman filter. Our scheme also features the capability to selectively update the state estimates as opposed to conventional techniques that require continual updating. More interestingly, simulations show meansquare error performance of the proposed algorithm to be almost identical to the Kalman filter, which is known to be optimal in Gaussian n...